# Solving for non moving points of a 1-D wave

gabbagabbahey
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If I gave you the function f(x,y)=3xy(x-7)(y+2)(x+2y), could you tell me which values of 'x' made that expression zero?

I would say 0, 7, and -2y if we knew what y is

but-2y would mean there's some kind of dependency between the two variables?

gabbagabbahey
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Yes, the roots x=0 and x=7 are independent of y, while the root x=-2y is not....the same ideas apply to the expression in post #18....

$\alpha=0$ is a root, and is independent of $\beta$ (and hence independent of $t$), and so values of $x$ where $\alpha=0$ will be stationary points (since the expression will be zero for all $t$ anytime $\alpha=0$).

There will also be six complex or real roots of the factor:

$(2087-16500 \alpha ^2+32928 \alpha ^4-18816 \alpha ^6-16500 \beta ^2+131760 \alpha ^2 \beta ^2$
$-263424 \alpha ^4 \beta ^2+150528 \alpha ^6 \beta ^2+32928 \beta ^4-263424 \alpha ^2 \beta ^4$
$+526848 \alpha ^4 \beta ^4-301056 \alpha ^6 \beta ^4-18816 \beta ^6+150528 \alpha ^2 \beta ^6-301056 \alpha ^4 \beta ^6+172032 \alpha ^6 \beta ^6)$

But they will all depend on $\beta$; and hence they also depend on $t$; and so they are not stationary points (stationary points are stationary for all $t$, not just specific values!).

So....the only stationary points are the points where $\alpha=0$

so to solve for the roots then, would just plain old factoring be the best way?

gabbagabbahey
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You mean solving for the 6 roots of that ugly expression in my last post?

yes heh, considering I don't know how to get maple to do anything correctly, what are some alternatives to find the roots? Thanks

gabbagabbahey
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Well, the expression In post #18 is fully factored....that means that the roots of the ugly factor are going to be difficult to find....BUT!!!!! you don't need to find them because they will all depend on $\beta$, whioch means they will only be roots for certain values of $t$, and as I said in post #28, that means that those roots are not stationary points....do you not understand this?

yes but...so am I suppose to say that even though they seem to be stationary, that's not the case since they will be dependent on beta?

gabbagabbahey
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Why do you say "they seem to be stationary"?

That's because we were to graph 10 plots at various t, I decidedly went t=1..10 and from the graphs the roots looked stationary

Also I had the impression that the problem would've been simpler in terms of all the trig stuff...

gabbagabbahey
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Oh.....try graphing it for t=0.2 and t=0.4...do they still look stationary?

It definitely does not. So I did a bad job in choosing the appropriate time interval then? My graphs definitely made me think I'd be getting answers for nonmoving x values...

regardless.... Thank you very much for being so patient. It makes sense how you approached this problem now. once again, Thanks!!!!!!!! :P